A \(2.0\,\mathrm{kg}\) cart moving at \(3.0\,\mathrm{m\,s^{-1}}\) sticks to a \(1.0\,\mathrm{kg}\) cart at rest. Find the common speed.

An isolated object initially at rest splits into two pieces. One piece has momentum \(5\hat{\imath}\,\mathrm{kg\,m\,s^{-1}}\). Find the other piece's momentum.

A \(0.010\,\mathrm{kg}\) pellet leaves a \(4.0\,\mathrm{kg}\) launcher at \(400\,\mathrm{m\,s^{-1}}\). Find the launcher's recoil speed, assuming the system starts at rest.

Two objects have momenta \(3\hat{\imath}+4\hat{\jmath}\) and \(-\hat{\imath}+2\hat{\jmath}\), in \(\mathrm{kg\,m\,s^{-1}}\). Find total momentum and its magnitude.

A \(60\,\mathrm{kg}\) skater at rest throws a \(5.0\,\mathrm{kg}\) package forward at \(6.0\,\mathrm{m\,s^{-1}}\). Find the skater's recoil speed.

A \(4.0\,\mathrm{kg}\) system initially has total momentum \(10\hat{\imath}\,\mathrm{kg\,m\,s^{-1}}\). An external impulse \(-6\hat{\imath}\,\mathrm{N\,s}\) acts. Find the final center-of-mass velocity.

Two carts at rest push apart on a frictionless track. Cart A has mass \(2.0\,\mathrm{kg}\), cart B has mass \(5.0\,\mathrm{kg}\), and cart A moves left at \(4.0\,\mathrm{m\,s^{-1}}\). Find cart B's velocity and explain why total momentum is conserved.

A \(1.5\,\mathrm{kg}\) object moving east at \(8.0\,\mathrm{m\,s^{-1}}\) explodes into two fragments. A \(0.50\,\mathrm{kg}\) fragment moves north at \(6.0\,\mathrm{m\,s^{-1}}\). Find the velocity vector of the other fragment.

A moving shell of mass \(6.0\,\mathrm{kg}\) travels east at \(12\,\mathrm{m\,s^{-1}}\) and explodes into three fragments. Two fragments have momenta \(20\hat{\imath}+15\hat{\jmath}\) and \(18\hat{\imath}-9\hat{\jmath}\), in \(\mathrm{kg\,m\,s^{-1}}\). Find the third fragment's momentum. State the isolation assumption and interpret the transverse components.

An isolated object of mass \(M\) moves with speed \(V\) in the \(x\)-direction and breaks into masses \(\alpha M\) and \((1-\alpha)M\), where \(0<\alpha<1\). The first fragment moves perpendicular to the original direction with speed \(u\). Derive the second fragment's velocity components and the extra kinetic energy created by the breakup. State assumptions.